Question

Explain for which of the following systems a factor of1/N! is needed in going from Q to q. a) CO2(g); b) graphite (s); c) diamond (s); and d) ice (s). (Ans. Just CO2(g))

Answer is provided, but need explanation

Answer 1

it is not clear what your Q and q are. But, assuming Q is a quantity(like energy E) describing the whole sytem and q describing one of it's components. ( And also guessing from the fact that 1/N! is usually the factor that differentiates the equations of a indistinguishable particles statistics and distinguishable particles statistics, I think the explanation is as follows).

Graphite,diamond and ice are crystals. So, the particles in the system can be distinguished by their positions in the crystal lattice. Therefore these 3 systems obey distinguishable particles statistics.

Whereas, In CO2, Since it is a gas you cannot distinguish between 1 gas molecule and the other (since it has only identical co2 molecules) these are indistinguishable particles. So, when you try to calculate the total energy by adding number of particles with a particular energy and so on(i.e, by counting the microstates)... we will end up overcounting. i.e, The system with 1st particle having E1 and 2nd particle having energy E2 is same as 1st particle having E2 and 2nd particle having E1 in the case of indistinguishable particles. The factor is then, as you say, calculated by permuting all identical particles in a each state (configuration). This is usually just N! for N identical particles. If there were coupling energies between more than 1 atom, this factor may change.